9/12/2023 0 Comments 9 dots 4 lines brain teaserFinally, various results for the a × b array of dots are proven. Moreover, the further constraint that the closed path remain within the convex hull of the array of dots can be satisfied for all n > 5. Golomb and John Selfridge showed that the unicursal polygonal path of 2 n - 2 segments exists on the n × n array for all n > 3 with the further constraint that the path be closed, i.e., it starts and ends at the same point. In 1956, the conjecture was proven by John Selfridge. Klamkin showed that if n > 2, then 2 n - 2 line segments are sufficient and conjectured that it is necessary too. Various such extensions were stated as puzzles by Dudeney and Loyd with different added constraints. If, instead of the 3-by-3 square lattice, we consider the n-by- n square lattice, then what is the least amount of lines needed to connect the dots without lifting the pen? Or, stated in mathematical terminology, what is the minimum- segment unicursal polygonal path covering the n × n array of dots? It is also possible to fold the paper flat, or to cut the paper into pieces and rearrange it, in such a way that the nine dots lie on a single line in the plane. Thus a single line can be drawn connecting all nine dots-which would appear as three lines in parallel on the paper, when flattened out. Īnother way to use only a single line involves rolling the paper into a three-dimensional cylinder, so that the dots align along a single helix (which, as a geodesic of the cylinder, could be considered to be in some sense a straight line). ![]() Or, if the line is allowed to be arbitrarily thick, then one line can cover all of the points. If the dots are assumed to have some finite size, rather than to be infinitesimally-small mathematical grid points, then it is possible to connect them with only three slightly-slanted lines. Various published solutions break the implicit rules of the puzzle in order to achieve a solution with even fewer than four lines. ![]() The inherent difficulty of the puzzle has been studied in experimental psychology. According to Daniel Kies, the puzzle seems hard because we commonly imagine a boundary around the edge of the dot array. The phrase thinking outside the box, used by management consultants in the 1970s and 1980s, is a restatement of the solution strategy. To do so, one goes outside the confines of the square area defined by the nine dots themselves. I haven’t been able to emulate this solution however the required folds can be viewed here.It is possible to mark off the nine dots in four lines. Below is a photo I took of this solution:Ī second one line solution involves origami and a specific placement and size of the dots on the page. The paper is then folded and a straight line is drawn connecting the dots. The method I came up with when trying to solve this puzzle involved tearing the paper, not so much that any piece comes loose. There are several one line solutions to this puzzle. The three-line solution can be viewed below: There are rotations, reflections and variants to this solution: The one line solution requires that the paper be folded. Scroll down for a clue and further down for the answer.Ĭlue: The lines drawn are permitted to leave the area of the dots and continue on into the blank portion of the page in order to gain a better angle for subsequent lines. Printing the dots out or drawing them on paper is required for this solution. Rather than using 4 lines, can you solve it with 3 straight continuous lines?įinally, for anyone brave (or foolish enough) there is a more challenging further question that requires just 1 straight line to connect the 9 dots. There is a more challenging version of this puzzle. The task commonly posed is to draw 4 straight lines without taking your pen off the paper so that all 9 dots are joined. ![]() This well-known problem entails 9 dots as seen below.
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